3 Answers
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A possible answer to part B of your question:
The first mishnah in Avodoh Zoroh says “For three days before the idolatrous festivals of the idolaters it is prohibited to have business dealings with them.” The reason, says Bartenuro, is in case they go to their Avodoh Zoroh on the festival and give thanks for the good deal.
It seems therefore that we must not cause the non-Jew to give thanks (and by extension, pray) to his Avodoh Zoroh.

What about Shituf? If he prays to the G-d of Abraham, Isaac and Jacob, and also his other deity that he either equates or conflates or puts on par with G-d (Has VeShalom), would the prayers to HaShem be overshadowed by the other thoughts in his head?
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Seth JSep 9 '11 at 15:34

I'm a ben Noach and was raised Catholic (and obviously left it). I'd just leave it be and only ask for prayers if the person is bnei Noach to some extent.

For instance, if they're religious in some other faith and pray, they'll just chalk it up to their belief system if the person gets better.

If they're bnei Noach then it is perfectly fine to pray for someone's healing as long as they refrain themselves from saying Jewish phrases in reference to themselves such as "asher kidshanu...etc", unless there is some bnei Noach prayer I don't know about that Noach was commanded to pray.

Just remember that all non-Jews don't follow the Sheva Mitzvot and could be in Avodah Zarah and then would such a blessing matter? It might even be harmful?

I would ask a question to the question. If asking for someone else to pray is this because the one asking is not sufficient? Was Abraham a Jew when he first prayed? What benefit is there in being a Jew if your prayers are not heard, and those prayers of a non-jew by birth are heard?
if I'm understanding the question right then I'd say that if the Jew by birth is having to ask another non-jew by birth to pray then that person needs to get back in a right relationship with G-d.

you're mistaking the issue. The problem is the question of where the individual is directing their prayers. Further, it has nothing to do with one's own prayers going unheard - rather, it's just a matter of there being added benefit in additional prayers.
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yoelJun 19 '12 at 19:31

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+1 to @yoel, but thank you, ironman99, for putting into perspective how one needs to remain focused on one's own prayers as much as asking for others to join.
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Seth JJun 19 '12 at 19:36

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@yoel, I see - thank you for the clarification. I've found that sometimes when in a situation asking those of no faith to pray generates in them a curiosity. It may be that G-d never opens their eyes to the truth, but it may open a door for you to tell them more about G-d. One of the saddest things that has happen in the United States - in my opinion - was for the School Corporations to prevent prayer in school. G-d may have only heard a few of those prayers in the 1st place, but the denial of prayer brought more victory for the evil one then prayers that were said but never heard.
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ironman99Jun 19 '12 at 19:47

@ who ever gave me a negative for my answer. It would be appreciated - especially when giving a negative - to produce some sort of educated disagreement with my answer. A negative should not be given just because of a disagreement of the answer given. So basically, please explain the reason for the negative.
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ironman99Jun 20 '12 at 14:23